Making of chaos
We could constitute the feedback loop for an artificial heart
automatic control algorithm by the use of an autonomic nerve discharges
or Mayer wave in peripheral vascular resistances, and or so. It may be
make a complicated structures in time series data as ﾒcomplex systemﾓ
like chaos or fractal.
However, is it really make complex system like circulatory regulatory system in creatures ?
A few investigators have challenged this problem. For example, Cavalcanti et al, reported a very interesting article in 1996 [20,21]. The profile was tried to be explained here. Model of baroreceptor reflex series was constituted with simple mathematics model. Simulation with various delay in the feed back loop were carried out. Behavior of the hemodynamics repeats bifurcation from the simple limit cycle vibration. Before long, it changes to the condition which is chaotic. With alteration only of delay, time series showed period doubling bifurcation, and reached to the chaos. This kind of experiment wasn't carried out at the past. As a result, interest is attracted very much.
If it is considered from a viewpoint of artificial heart control, baroreflex delay means the control delay of an artificial heart. Of course, hemodynamics is a signal of time series such as blood pressure or heart rate. We tried to generate deterministic chaos in the feedback loop model of an artificial heart control.
Section equivalent to cardiovascular system had been simulated with the three elements Windkessel represented with simple electric circuit. Three elements Windkessel is expressed as the systemic vascular resistance (R), arterial compliance (C) and the characteristic impedance (r). In this model, input is aortic flow, and the output is aortic pressure. In other words, it is been similar to cardiovascular system with electric circuit model. Formula formed with this model is (4) and (5). Each value of these three elements was adopted from the data mentioned in the literature of the physiology.
dP(t) = ωｔ[ RQ(t) - Ps(t)], ωｔ = 1 / RC (4)
P(t) = Ps + Q(t) (5)
In this simulation, non-linear curve was used for determination. Cardiac cycle (T) and Stroke volume (SV) is decided from arterial blood pressure (P), which is an data of a baroreflex. In this study, non-linear curve as the sigmoid function was used for that purpose as we shown in (6) and (7). Concerning the each constant, we sought the value from the literature in physiology. Figure 14 and 15 showed its relationship.
T(P) = Ts + (Tm - Ts) / ( 1 + γe -αP/Pn) (6)
SV(P) = Svmax / ( 1 + β( P / Pv - 1 )-k) (7)
Delay t was introduced in the feedback loop in this study. After the delay time r, it was input into the next element. This was included in the feed-back loop branch. This time delay r was altered as a parameter, when we carried out the simulation.
Figure 17 showed simulated time series data of blood pressure, RR interval and stroke volume with time delay 0.6 s. Right side showed reconstructed attractor of the simulated time series with BP, HR, and SV. In the time series data of these parameters, all signals showed the damping oscillation. The whorl which converges in a point was shown in the reconstructed three dimensional attractor.
Figure 18 showed simulated time series data with time delay 1.5 s. In the time series data of these parameters, all signals showed the constant periodic oscillation. In the reconstructed three dimensional attractor, the orbit converged and drew circle. This showed so called the limit cycle attractor.
A little complex result with time delay 1.8 s was shown in figure 19, simulated cardiovascular signals showed interesting time series suggesting one set periodic oscillation with two mountains. When we reconstructed these data into the three dimensional map, the orbit converged and becomes track with two rounds.
And complex time series was shown with time delay 2.5 s, as we shown in figure 20. Simulated cardiovascular signal showed complex oscillation without any rule. We tried to embed these time series into the three dimensions. As we shown in right side of figure, the orbit which isn't accompanied with convergence is drawn.
We paid attention to the behavior with time delay 2.5 s. Time series data and the power spectrum analysis of simulated heart rate variabilities were shown in figure 9.
Time series wave form of HR becomes complex oscillation same as other cardiovascular signal, too. There is frequency spectrum as the biggest beak approximately 0.12 Hz with many mountains.
We tried to let initial value change at the time of delay 2.5 seconds inconsiderably. As we shown in fig.10, a figure repeated time series wave form of each HR, and it was written. A result draws time series wave form of oscillation different at all mutually.
There is a point showing the characteristics, which showed deterministic chaos, when tries to pay attention to this condition. At first, frequency spectrum of the simulated heart rate variability showed it. As we shown in the figure, power spectrum has a peak on the section which wasn't harmonics composition. And a lot of another small mountains were found. These many spectrums was considered to have made a complex signal. This spectral pattern was, of course, one of the characteristics of the deterministic chaos. Only harmonics composition shows a peak in case of limit cycle.
The second was behavior of the simulated data at having let initial value alter. As one of fundamental characteristics of chaos, there was the sensitive initial value dependence. Actually it was clear if watched a figure. We had tried to let alter the initial value a little and did simulation. Time series wave form showed oscillation different at all.
This phenomenon backed up the existence of the deterministic chaos.
When a spectrum was tried to be paid attention to, a peak having the biggest power in 0.12 Hz is found. This was, of course, a peak equivalent to Mayer wave. Fluctuation of period for approximately 10 seconds was the character which was very special in the creatures. By the use of model with this simple electrical circuit, this interesting fluctuation was caused. In this simulation, we might be able to consider that the delay would cause this phenomenon.
Non-linear curve, which was, of course, common in creatures, was tries to be considered this time. From blood pressure, determination of cardiac cycle and SV was performed with non-linear curve of sigmoid style. From these two curves, the curve which decided flow from blood pressure was drawn. In this simulation, operating range of blood pressure was from 70 to 100 mmHg, which was similar to human body. The section of graph had tried to be paid attention to. In this curve, this range had drew curve of the convexity which just resembled a logistic function. With this cause, ﾒstretchingﾓ and ﾒholdingﾓ might be happened. And it is considered to have caused the condition which was chaotic.
However, of course, this electrical simulation model still includes some problems, yet. Some important problems were shown below.
1. Complex cardiovascular system was simulated with simple electric circuit model.
2. Stroke volume (SV) was decided with arterial blood pressure.
3, Simulation only of the systemic circulation by left heart
It was thought that the first problems may be important. We need to consider whether this model was able to simulate the original living body and which degree. Model of baroreflex series is made from real data of the creatures, and further simulation needs to be retried in future. And SV depends on pressure of the atrium which is pre-load. In electrical simulation model of this study, it is decided only by the arterial blood pressure, which is afterload. And, of course, behavior of the pulmonary circulation was important. Accordingly this baroreceptor reflex model is not enough, yet. Baroreceptor reflex system of the original living body will not be expressed in only this simple electrical simulation model.
However, it is considered that chaotic dynamics may be caused with simple three elements circuit with some delay. These results suggest that we could achieve the chaotic dynamics like human body by the use of feed back automatic control algorithm of an artificial heart system. These results may be very important in future, if we consider the Age, in which Artificial organs become very common, and QOL of Artificial organs become important. So, studies will want to be continued with more refined procedure.
However, is it really make complex system like circulatory regulatory system in creatures ?
A few investigators have challenged this problem. For example, Cavalcanti et al, reported a very interesting article in 1996 [20,21]. The profile was tried to be explained here. Model of baroreceptor reflex series was constituted with simple mathematics model. Simulation with various delay in the feed back loop were carried out. Behavior of the hemodynamics repeats bifurcation from the simple limit cycle vibration. Before long, it changes to the condition which is chaotic. With alteration only of delay, time series showed period doubling bifurcation, and reached to the chaos. This kind of experiment wasn't carried out at the past. As a result, interest is attracted very much.
If it is considered from a viewpoint of artificial heart control, baroreflex delay means the control delay of an artificial heart. Of course, hemodynamics is a signal of time series such as blood pressure or heart rate. We tried to generate deterministic chaos in the feedback loop model of an artificial heart control.
Section equivalent to cardiovascular system had been simulated with the three elements Windkessel represented with simple electric circuit. Three elements Windkessel is expressed as the systemic vascular resistance (R), arterial compliance (C) and the characteristic impedance (r). In this model, input is aortic flow, and the output is aortic pressure. In other words, it is been similar to cardiovascular system with electric circuit model. Formula formed with this model is (4) and (5). Each value of these three elements was adopted from the data mentioned in the literature of the physiology.
dP(t) = ωｔ[ RQ(t) - Ps(t)], ωｔ = 1 / RC (4)
P(t) = Ps + Q(t) (5)
In this simulation, non-linear curve was used for determination. Cardiac cycle (T) and Stroke volume (SV) is decided from arterial blood pressure (P), which is an data of a baroreflex. In this study, non-linear curve as the sigmoid function was used for that purpose as we shown in (6) and (7). Concerning the each constant, we sought the value from the literature in physiology. Figure 14 and 15 showed its relationship.
T(P) = Ts + (Tm - Ts) / ( 1 + γe -αP/Pn) (6)
SV(P) = Svmax / ( 1 + β( P / Pv - 1 )-k) (7)
Delay t was introduced in the feedback loop in this study. After the delay time r, it was input into the next element. This was included in the feed-back loop branch. This time delay r was altered as a parameter, when we carried out the simulation.
The simulation model.
In this study, deterministic chaos was tried to be caused only with alteration of delay by the use of this very simple model. Block diagram was made like a fig.16. Simulation was carried out about this model. Time series data of the simulated hemodynamic parameters, three dimensional attractors and frequency spectrum was observed here. We had used MATLAB for simulation and analysis.Experimental result
Behavior of the hemodynamic parameters were simulated in the simple electrical circuit model with a changes of the delay time r. Various interesting phenomenon were shown in the simulated data. Only some typical data were picked up and shown in the figures.Figure 17 showed simulated time series data of blood pressure, RR interval and stroke volume with time delay 0.6 s. Right side showed reconstructed attractor of the simulated time series with BP, HR, and SV. In the time series data of these parameters, all signals showed the damping oscillation. The whorl which converges in a point was shown in the reconstructed three dimensional attractor.
Figure 18 showed simulated time series data with time delay 1.5 s. In the time series data of these parameters, all signals showed the constant periodic oscillation. In the reconstructed three dimensional attractor, the orbit converged and drew circle. This showed so called the limit cycle attractor.
A little complex result with time delay 1.8 s was shown in figure 19, simulated cardiovascular signals showed interesting time series suggesting one set periodic oscillation with two mountains. When we reconstructed these data into the three dimensional map, the orbit converged and becomes track with two rounds.
And complex time series was shown with time delay 2.5 s, as we shown in figure 20. Simulated cardiovascular signal showed complex oscillation without any rule. We tried to embed these time series into the three dimensions. As we shown in right side of figure, the orbit which isn't accompanied with convergence is drawn.
We paid attention to the behavior with time delay 2.5 s. Time series data and the power spectrum analysis of simulated heart rate variabilities were shown in figure 9.
Time series wave form of HR becomes complex oscillation same as other cardiovascular signal, too. There is frequency spectrum as the biggest beak approximately 0.12 Hz with many mountains.
We tried to let initial value change at the time of delay 2.5 seconds inconsiderably. As we shown in fig.10, a figure repeated time series wave form of each HR, and it was written. A result draws time series wave form of oscillation different at all mutually.
Discussion
By the use of information of an autonomic nervous system like sympathetic nerve activity of the Mayer wave in the peripheral vascular resistances, we can make feed back loop for an artificial heart automatic control system. In this study, we make simulation model by the use of the electrical circuit model. Behavior changes in the simulation model altering delay time could be shown in figures. When delay in simulation was small comparatively, a signal of everything showed damping oscillation. The whorl which converges in a point was shown in the reconstructed three dimensional attractor. And if we set up larger delay in the simulation circuit, periodic oscillation, and then, period doubling phenomenon were shown in the reconstructed attractor. After the repeating of the period doubling according to the increase of the delay time, we could get strange attractor suggesting the existence of deterministic chaos, when we set up the delay time of 2.5 seconds in this simulation model electrical circuit. This complex oscillation didn't disappear even if delay was set up more than 2.5 seconds.There is a point showing the characteristics, which showed deterministic chaos, when tries to pay attention to this condition. At first, frequency spectrum of the simulated heart rate variability showed it. As we shown in the figure, power spectrum has a peak on the section which wasn't harmonics composition. And a lot of another small mountains were found. These many spectrums was considered to have made a complex signal. This spectral pattern was, of course, one of the characteristics of the deterministic chaos. Only harmonics composition shows a peak in case of limit cycle.
The second was behavior of the simulated data at having let initial value alter. As one of fundamental characteristics of chaos, there was the sensitive initial value dependence. Actually it was clear if watched a figure. We had tried to let alter the initial value a little and did simulation. Time series wave form showed oscillation different at all.
This phenomenon backed up the existence of the deterministic chaos.
When a spectrum was tried to be paid attention to, a peak having the biggest power in 0.12 Hz is found. This was, of course, a peak equivalent to Mayer wave. Fluctuation of period for approximately 10 seconds was the character which was very special in the creatures. By the use of model with this simple electrical circuit, this interesting fluctuation was caused. In this simulation, we might be able to consider that the delay would cause this phenomenon.
Non-linear curve, which was, of course, common in creatures, was tries to be considered this time. From blood pressure, determination of cardiac cycle and SV was performed with non-linear curve of sigmoid style. From these two curves, the curve which decided flow from blood pressure was drawn. In this simulation, operating range of blood pressure was from 70 to 100 mmHg, which was similar to human body. The section of graph had tried to be paid attention to. In this curve, this range had drew curve of the convexity which just resembled a logistic function. With this cause, ﾒstretchingﾓ and ﾒholdingﾓ might be happened. And it is considered to have caused the condition which was chaotic.
However, of course, this electrical simulation model still includes some problems, yet. Some important problems were shown below.
1. Complex cardiovascular system was simulated with simple electric circuit model.
2. Stroke volume (SV) was decided with arterial blood pressure.
3, Simulation only of the systemic circulation by left heart
It was thought that the first problems may be important. We need to consider whether this model was able to simulate the original living body and which degree. Model of baroreflex series is made from real data of the creatures, and further simulation needs to be retried in future. And SV depends on pressure of the atrium which is pre-load. In electrical simulation model of this study, it is decided only by the arterial blood pressure, which is afterload. And, of course, behavior of the pulmonary circulation was important. Accordingly this baroreceptor reflex model is not enough, yet. Baroreceptor reflex system of the original living body will not be expressed in only this simple electrical simulation model.
However, it is considered that chaotic dynamics may be caused with simple three elements circuit with some delay. These results suggest that we could achieve the chaotic dynamics like human body by the use of feed back automatic control algorithm of an artificial heart system. These results may be very important in future, if we consider the Age, in which Artificial organs become very common, and QOL of Artificial organs become important. So, studies will want to be continued with more refined procedure.
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